Linked Questions

61
votes
8answers
7k views

Using Gröbner bases for solving polynomial equations

In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean ...
10
votes
6answers
473 views

Given $x+y$ and $x\cdot y$, what is $x^3+ y^3$ ?

I have been looking at an assortment of high school number sense tests and I noticed a reoccurring problem that states what x+y is and what $x\cdot y$ is then asks for $x^3+ y^3$. I want to know how ...
12
votes
4answers
4k views

Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer?

Are there any computer algebra systems with the functionality to allow me to enter in an explicit symmetric polynomial and have it return that polynomial in terms of the elementary symmetric ...
0
votes
5answers
5k views

If $x + \frac{1}{x} = \sqrt{3}$, then find $x^{18}$ [duplicate]

If $x + \frac{1}{x} = \sqrt{3}$, then find $x^{18}$. This is sorta like If $x^3+\frac{1}{x^3}=18\sqrt{3}$ then to prove $x=\sqrt{3}+\sqrt{2}$. So my question is, do I solve my question the same way as ...
4
votes
3answers
419 views

Relation between roots and coefficients - manipulation of identities

The polynomial $x^3+3x^2-2x+1$ has roots $\alpha, \beta, \gamma$ . Find $$\alpha^2(\beta + \gamma) + \beta^2(\alpha + \gamma) + \gamma^2(\alpha + \beta)$$ I tried finding the relation using $-b/a$, $...
0
votes
3answers
176 views

With a product and sum of $x$ and $y$, calculate $9x^2+15y^2$

If We have $x+y=4 $, $ x\cdot y=-1$ and $x>y$ than $9 x^2+15 y^2=\;\;? $
2
votes
3answers
146 views

Find the value of equation?

Suppose \begin{align*} a+b+c &= 20\\ \frac 1a + \frac 1b + \frac 1c &= 30 \end{align*} Then find the value of $$ \frac ab + \frac ba + \frac ac + \frac ca + \frac bc + \frac cb $$ ...
0
votes
2answers
462 views

Quadratic polynomial over $K$

Let $K$ be any subfield of $\mathbb{C}$ and let $m(t)$ be a quadratic polynomial over $K$. show that all zeros of $m(t)$ lie in an extension $K(\alpha)$ of $K$ where $\alpha^2=k\in K$. Thus allowing ...
1
vote
2answers
273 views

Find a polynomial only from its roots

Given $\alpha,\,\beta,\,\gamma$ three roots of $g(x)\in\mathbb Q[x]$, a monic polynomial of degree $3$. We know that $\alpha+\beta+\gamma=0$, $\alpha^2+\beta^2+\gamma^2=2009$ and $\alpha\,\beta\,\...
2
votes
1answer
264 views

Please help explain how “Any symmetric sum can be written as a Polynomial of the elementary symmetric sum functions”

I was browsing through the Art Of Problem Solving website and came across this: "Any symmetric sum can be written as a Polynomial of the elementary symmetric sum functions, for example $x^3 + y^3 + ...
2
votes
2answers
118 views

How prove $\frac{(a-b)^4+(b-c)^4+(c-a)^4}{(a+b+c)^4}=2$

Let $a,b,c\in R$,and such $ab+bc+ac=0,a+b+c\neq 0$ show that $$\dfrac{(a-b)^4+(b-c)^4+(c-a)^4}{(a+b+c)^4}=2$$
5
votes
1answer
133 views

Prove the product of a polynomial function of the roots of another polynomial is an integer.

I noticed this while solving another problem on this site. Let $P(x)$ be a polynomial in $x$ with integer coefficients, and let the roots of $P(x)=0$ be $r_1, r_2 \ldots ,r_n$, where multiple $r_i$ ...
0
votes
2answers
237 views

Symmetric roots of polynomial

Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the polynomial $x^3 - x^2 + 2x - 3$ $\in \mathbb{C}[x]$. Calculate $\alpha_1^3 + \alpha_2^3 + \alpha_3^3$. What to do here exactly? I already ...
0
votes
1answer
135 views

$(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ in terms of elementary symmetric polynomials?

Is it possible to express $(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ in terms of elementary symmetric polynomials ? What I tried was ...
0
votes
2answers
108 views

Symmetric polynomial written in elementary polynomials

Consider the symmetric polynomial in three variables $x, y, z$ $$|x^2y+y^2z+z^2x-xy^2-yz^2-zx^2|.$$ A Theorem says that it can be written in elementary symmetric polynomials $$s_1=x+y+z, \quad s_2=...

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